Download Aggregation Functions: A Guide for Practitioners by Gleb Beliakov PDF

By Gleb Beliakov

Aggregation of data is of basic significance within the development of information dependent structures in a number of domain names, starting from medication, economics, and engineering to decision-making procedures, synthetic intelligence, robotics, and desktop studying. This booklet provides a extensive advent into the subject of aggregation services, and gives a concise account of the homes and the most periods of such features, together with classical capability, medians, ordered weighted averaging capabilities, Choquet and Sugeno integrals, triangular norms, conorms and copulas, uninorms, nullnorms, and symmetric sums. It additionally offers a few cutting-edge recommendations, many graphical illustrations and new interpolatory aggregation capabilities. a specific cognizance is paid to id and development of aggregation services from program particular standards and empirical information. This publication offers scientists, IT experts and approach architects with a self-contained easy-to-use advisor, in addition to examples of computing device code and a software program package deal. it's going to facilitate building of determination aid, specialist, recommender, regulate and plenty of different clever systems.

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Extra resources for Aggregation Functions: A Guide for Practitioners

Example text

N}. e. v(S) ≤ v(T ) whenever S ⊆ T ) and satisfies v(∅) = 0, v(N ) = 1. 74 (Choquet integral). 9) i=1 where x = (x(1) , x(2) , . . , x(n) ) is a non-decreasing permutation of the input x, and x(n+1) = ∞ by convention. By rearranging the terms of the sum, Eq. 9) can also be written as n x(i) − x(i−1) v(Hi ). 10) i=1 where x(0) = 0 by convention, and Hi = {(i), . . , (n)} is the subset of indices of n − i + 1 largest components of x. The class of Choquet integrals includes weighted arithmetic means and OWA functions as special cases.

N (xn ))) is called the dual of f with respect to N , or, for short, the N -dual of f . When using the standard negation, fd is given by fd (x1 , . . , xn ) = 1 − f (1 − x1 , . . , 1 − xn ) and we will simply say that fd is the dual of f . It is evident that the dual of a conjunctive aggregation function is disjunctive, and vice versa, regardless of what strong negation is used. Some functions are self-dual. 55 (Self-dual aggregation function). Given a strong negation N , an aggregation function f is self-dual with respect to N (for short, N -self-dual or N -invariant), if f (x) = N (f (N (x))), where N (x) = (N (x1 ), .

18. We denote by x the vector obtained from x by arranging its components in non-decreasing order, that is, x = xP where P is the permutation such that xP (1) ≤ xP (2) ≤ . . ≤ xP (n) . Similarly, we denote by x the vector obtained from x by arranging its components in non-increasing order, that is x = xP where P is the permutation such that xP (1) ≥ xP (2) ≥ . . ≥ xP (n) . 8 It is frequently interpreted as anonymity criterion: anonymous ballot papers can be counted in any order. 19. In fuzzy sets literature, the notation x() = (x(1) , .

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